Abstract

Euclidean rotations in \(\mathbb {R}^n\) are bijective and isometric maps. Nevertheless, they lose these properties when digitized in \(\mathbb {Z}^n\). For \(n=2\), the subset of bijective digitized rotations has been described explicitly by Nouvel and Rémila and more recently by Roussillon and Cœurjolly. In the case of 3D digitized rotations, the same characterization has remained an open problem. In this article, we propose an algorithm for certifying the bijectivity of 3D digitized rational rotations using the arithmetic properties of the Lipschitz quaternions.

Notes

Acknowledgments

The authors express their thanks to Éric Andres of Université de Poitiers for his very helpful feedback and comments which allowed us to improve the article.

The research leading to these results has received funding from the Programme d’Investissements d’Avenir (LabEx Bézout, ANR-10-LABX-58).